qualitative modelling and analysis of photosystem iiceur-ws.org/vol-988/paper3.pdfqualitative...

Qualitative modelling and analysis of Photosystem II?

Lubos Brim, Vilem Ded and David Safranek

Systems Biology Laboratory, Faculty of Informatics, Masaryk University, Botanicka 68a,602 00 Brno, Czech Republic

{brim,xded,xsafran1}@fi.muni.cz

Abstract. In this paper, a work-in-progress aiming at qualitative modelling ofphotosynthesis at the mechanistic cellular level by means of Petri nets is de-scribed. Presented preliminary results concentrate on modelling and analysis ofphotosystem II, a crucial component of photosynthesis. By employing qualitativemodel checking combined with invariant analysis we obtain new insights intoelectron transfer mechanisms studied in photosystem II.

1 Introduction

Photosynthetic reactions of green plants and some bacteria take place on a membraneof specialized organelles called thylakoids where a lot of protein complexes reside.The most important are photosystem I, photosystem II and cytochrome complex b6 f .These complexes are responsible for absorbing light and they bind the light energy thatis further used and transformed in later phases of photosynthesis. Products of thesereactions are oxygen and energy bound in ATP and NADPH. The first complex inthe whole reaction cascade is photosystem II which is responsible for absorbing light,splitting molecule of water and exciting electrons that further reduce subsequent com-plexes. [15]. Photosystem II is also considered to have main influence to measured data(especially light-induced fluorescence)[9].

Photosynthesis is often modelled using photochemical and redox reactions reflect-ing only some of measurable values, e.g., chlorophyll fluorescence, concentration ofcarbon dioxide or oxygen [15]. In the case of chlorophyll, fluorescence absorbance/e-mission is driven by femtosecond-scale reactions considered to be transitions of elec-trons. For these reactions, determination of kinetic rates is very difficult even impossi-ble. Moreover, the reactions are influenced by many other factors such as value of pHor environment temperature.

Recently, some estimating methods have been already used for creation of quanti-tative models of photosystem II using differential equations [11]. There are also manystudies trying to understand precise function of each component of photosystem II. Themeasured data are moreover commonly explained by very different theories [9]. Somemodels were analysed qualitatively [10], nevertheless, their development has been stillbased on incomplete quantitative knowledge.

? This work has been partially supported by the Czech Science Foundation grantNo. GAP202/11/0312. D. Safranek has been supported by EC OP project No.CZ.1.07/2.3.00/20.0256.

G. Balbo and M. Heiner (Eds.): BioPPN 2013, a satellite event of PETRI NETS 2013, CEUR Workshop Proceedings Vol. 988, 2013.

18 Brim, Ded, Safranek

The potential of using qualitative modelling in system biology is commonly under-rated because of the high level of abstraction. On the other hand, it has been shownmany times that qualitative modelling is very powerful and contributive in cases of lackor incompleteness of data [21].

Petri nets are very powerful formalism which has been proved useful for modellingof complex biological systems and for providing some interesting hypothesis about theirbehaviour only on bases of knowledge of interactions between its components. Notableexamples include discovery of various metabolite functions and components with veryspecific behaviour, e.g., cycling or accumulation [18]. For example, Petri nets have beensuccessfully applied to models of metabolic cascade [6–8]. Though based on a purelyqualitative model, the analyses have brought new insights. Other applications show theadvantages of using Petri nets for high level modelling of multi-cellular organisms [3].

To the best of our knowledge, there exist only a few applications of Petri nets tomodelling of photosynthesis. A general photosynthesis reaction is used as a simpleexample in [16] while a more serious case study is provided in [14] targeting non-photochemical quenching by means of metabolic P systems.

This paper describes a work-in-progress aiming at qualitative modelling of photo-synthesis at the mechanistic cellular level by means of Petri nets. Our first results con-centrate on modelling and analysis of photosystem II, a crucial component of photosyn-thesis. For model development and analysis, we have employed the tools Snoopy [19]and Charlie [5]. We show that qualitative model checking is able to verify some com-monly accepted theories about photosystem II.

2 Background and Problem Formulation

Fig. 1: Structure of photosystem II. Taken from [12].

Proc. BioPPN 2013, a satellite event of PETRI NETS 2013

Qualitative modelling and analysis of Photosystem II 19

Photosystem II can be divided into reaction center, oxygen evolving complex (OEC)and light harvesting complex (LHC). Reaction center is composed of cytochrome b559,chlorin complex P680 and two other proteins very similar to each other(D1 and D2).Each of them binds other redox active components: pheophytin (Pheo), quinone (Qafor D1 , Qb for D2), chlorophyll (Chlz), carotene (Car) and tyrosine (YZ for D1, YD forD1) [17, 20]. Because literature does not describe precise transitions inside complexP680, in this paper we neglect the fact that this complex consists of more than a singlemolecule. Very important is oxygen-evolving complex (OEC) bounded to P680. Thiscomplex is responsible for splitting the molecule of water and can be non-functionalunder some conditions [22].

The photosynthetic process begins with impact of photon to the LHC. Its energy istransited to the reaction center where it causes excitation of electron [13].

2.1 Basic electron path

The whole electron path begins with oxidation of complex of chlorins P680. Its electronreduces pheophytin Pheo. Then the electron continues to quinone QA and finally toquinone QB which is capable of carrying two electrons [20]. After receiving a secondelectron, quinone QB (now QB

-2) is now neutralized by two hydrogen cations from thestroma (the outside of thylakoid) to quinol QB

-2H2 and leaves PSII into PQ-pool(spacebetween PSII and complex b6 f containing 7-10 molecules of quinone/quinol). Into itsplace there comes a new neutral molecule of quinone QB. After leaving PSII, quinoltransports electrons to the next protein complex (b6 f ) and releases hydrogen into thethylakoid inner space turning again into quinone that is able to recycle back to PSII [9].

Fig. 2: Possible electron paths in Photosystem II.

When the P680 complex is oxidised, it receives one electron from tyrosine YZ (pro-tein is situated between OEC and P680) which is subsequently reduced by one of thefour electrons of OEC. After the fourth excitation of the electron from P680, the lastelectron is taken from OEC which is now capable to split two molecules of water. Thisreaction provides needed electrons and releases four hydrogen cations and two atomsof oxygen [20].



2.2 Alternative electron paths

Literature presents also other possible transitions of electron. In particular, there arepossibilities of reverse transitions. Existence of those transitions is known betweenmolecules of quinone QA and QB[11], Pheo and P680[23], QA and Pheo, P680 andYD, and finally YD and OEC[17]. There might be also a reverse reaction of binding(unbinding) of quinone QB(quinol QB

-2H2) [11].Except for those reverse transitions, literature describes also another direct transi-

tion of electrons. There is a possible transition from reduced QA-1 to oxidised P680 [20].

Other path is reduction of oxidised P680 by an electron from cyt b559 which then canreceive an electron from a molecule of Qb reduced at least once [20]. Oxidised cyt b559can receive an electron also from reduced pheophytin Pheo [1]. Such paths through cytb559 are important during inhibition of OEC function (e.g., by low temperature or dam-age). As long as P680 is oxidised, it becomes a strong oxidation agent and may causedestruction of the whole system. The alternative path through cyt b559 prevents suchevents by periodic reduction of P680 [1, 22]. Finally, another possible transitions areoxidation of tyrosine YD (a protein analogous to YZ bound to D2) and reduction of ox-idised P680 where YD serves as the electron donor only once and during permanentillumination it stays in the oxidised state [22].

2.3 Existing models of photosystem II

It is worth noting that the literature shows only model structures created to better un-derstand its content or those derived from quantitative models. There exist more models(described at structural level) differing in composition and interpretation of componentsand related transitions. This is the case mainly for OEC, pheophytin, cytochrome b559,and both molecules of tyrosine. OEC is modelled by four transitions between statesSi where i ∈ {0,1,2,3} refers to number of missing electrons. In some cases, a freeelectron state S4 is used which is connected to S0 signifying the split of a molecule ofwater.

Structures are always described by a network of electron transitions, state transitionsof a system, or by a reaction network of individual subunits.

Combination of these approaches was presented by Zhu et al. [23]. He dividedphotosystem into two subunits (P680/Pheo a QA/QB) and presented their state graphs.Moreover, his model contains OEC with Si states and a molecule of tyrosine. All men-tioned parts are connected by electron transitions.

Nedbal [15] showed very comprehensive structure composed by many state graphsof the subunit YZ /P680/ChlD/Pheo/QA. These graphs were connected by transitions re-ferring to change of four states of QB (QB, QB

-1, QB-2 and E representing absence of

non-presence of QB). Finally, these graphs were obtained by interconnecting the sub-graphs with reactions representing changes among S i states of OEC.

Lazar [10] discussed several structures of quantitative models and the possibility ofusing first-order and second-order kinetics for modelling reactions of (un)binding QBand reactions between OEC and subunit P680/QA/QB. Subsequent analysis of thesetwo approaches was focused mainly on quantitative features.

Other important model structures can be found also in [2, 20].



2.4 Problem specification

As mentioned above, existing models differ in the level of detail or in the modelling ap-proach employed. Naturally, there is an inevitable need to unify the notation of models.There are also many biological questions to be solved. For example:

– Does the behaviour and features of models depend on presence/absence of particu-lar photosystem II subunits?

– Are all the current theories about photosystem II valid?– Can the model reach some final (stable) state?– Are there any possible electron cycles which have not yet been observed?– Are the existing models correct?

3 Results

For answering these questions we need to create a bunch of models reflecting the ab-sence/presence of particular components. We perform static analysis (discovering in-variants and their interpretation) and dynamic analysis using computational tree logic(CTL).

3.1 Model development

There exist several approaches to modeling photosystem II using Petri nets dependingon interpretation of places and transitions. Some of them can be seen in Fig. 3. In thispaper, we employ places to represent states of observed components of photosystemII and transitions to represent electron transitions. Only exception is the case of QBwhere places refer to states of binding site or bounded molecule of QB and transitionsrepresent electron transfer (QA ↔ QB

-, QB- ↔ QB

-2) or (un)binding of QB or QB-2.

We consider a basic model including only irreversible reactions and quinone QBmodeled without E state. Refining the model by all possible states and related reactionsresults in a combinatorial explosion. Number of possible models created on a basis ofknowledge mentioned in Sec. 2 is 4320. For this work, we have selected 11 models:

A is basic model of PSII (see Fig. 4). It does not contain any irreversible reaction whileit contains only complex P680, QA and QB without E state. This model reflects onlythe basic forward path of electron as mentioned in [20].

B is an extension of the model A. There are considered molecules of pheophytin andtyrosine in the basic path. Moreover, QB is modelled with state E.

C represents the model B extended by the reverse electron flow from quinone QA toP680.

D represents model B extended by molecule of tyrosine YD.E refers to model C extended by molecule of cytochrome B599 and related transi-

tions.F is a model created by the union of models E and D. It makes the complete model

without any reversible transitions.



Fig. 3: The figure shows different approaches to modelling of photosystem II using Petrinets. In (a), places represent the component itself. In this case, the maximal number oftokens per place must be specified (normally it is 1 but QB can carry 2 and OEC up to 4electrons). In (b), places represent states of individual components. Model c is createdas combination of models a and b.

G represents model B extended by reverse transitions between YZ and P680, P680 andPheo, Pheo and QA.

H represents model F extended by reverse transitions between QA and QB.I is a model created by the union of models D and H.J refers to model H extended by reverse transitions between states QB

-2 and E (E andQB) which represents possible reverse binding of QB

-2 (reverse unbinding of QB).K is the complete and the most complex model containing all components and transi-

tions mentioned in literature (see Fig. 4).

Moreover, we have created all models in two forms. Variant (a) (e.g., Ka) denotesmodels containing OEC. Models marked (b) refer to the system with a non-functionalOEC.



Fig. 4: The figure shows the simplest (a) and the most complex (b) model with therespective initial marking.

3.2 Initial marking construction

Before the analysis we need to specify what is the initial state (marking) of our sys-tem. It would not be necessary for models with the state space making a single stronglyconnected component. In this case, all states are reachable regardless of current state ofmodel. Nevertheless, there exist models that do not satisfy that. The connected compo-nent in the state space can be interpreted as a permanent outflow/inflow of the electronfrom/into some system subunit. For the whole analysis we expect all components inneutral state and OEC in the state S0 (with four electrons present). Employing theseconstraints, the initial marking M0 is the following:

P680 P680+ Pheo Pheo- QA QA- QB QB

- QB-2 E YZ YZ

- S0 S1 S2 S3 YD Yz- B559 B559

-

1 0 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0



3.3 Invariants

P-invariants determine sets of places where the total number of tokens is conserved.For our models this can be interpreted as preserving all of redox active components inthe model, e.g., P-invariants show that all components were present in the model duringthe process execution.

t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12

P → Pheo 1 0 0 0 0 0 0 1 1 1 1 4Pheo → P 1 0 0 0 0 0 0 0 0 0 0 0

Pheo → QA 0 1 0 0 0 0 0 0 1 1 1 4QA → Pheo 0 1 0 0 0 0 0 0 0 0 0 0QA → QB 0 0 1 0 0 0 0 0 0 1 0 2QA → QB

- 0 0 0 1 0 0 0 0 0 0 1 0QB

- → QA 0 0 1 0 0 0 0 0 0 0 0 0QB

-2 → QA 0 0 0 1 0 0 0 0 0 0 0 0QB

-2 → E 0 0 0 0 0 1 0 0 0 0 0 2E → QB

-2 0 0 0 0 0 1 0 0 0 0 0 0E → QB 0 0 0 0 0 0 1 0 0 0 0 2QB → E 0 0 0 0 0 0 1 0 0 0 0 0YZ → P 0 0 0 0 1 0 0 0 0 0 0 4P → YZ 0 0 0 0 1 0 0 0 0 0 0 0S0 → YZ 0 0 0 0 0 0 0 0 0 0 0 1S1 → YZ 0 0 0 0 0 0 0 0 0 0 0 1S2 → YZ 0 0 0 0 0 0 0 0 0 0 0 1S3 → YZ 0 0 0 0 0 0 0 0 0 0 0 1YD → P 0 0 0 0 0 0 0 0 0 0 0 0QA → P 0 0 0 0 0 0 0 0 1 0 0 0

Pheo → B559+ 0 0 0 0 0 0 0 1 0 0 0 0QB

- → B559+ 0 0 0 0 0 0 0 0 0 1 0 0QB

-2 → B559+ 0 0 0 0 0 0 0 0 0 0 1 0B559 → P 0 0 0 0 0 0 0 1 0 1 1 0

Table 1: Table shows which T-invariants were (1) or were not (0)found in particular models. All relatedmodels of model group b contain thesame invariants with the only exceptionof t12.

t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12

A 0 0 0 0 0 0 0 0 0 0 0 1B 0 0 0 0 0 0 0 0 0 0 0 1C 0 0 0 0 0 0 0 0 1 0 0 1D 0 0 0 0 0 0 0 0 0 0 0 1E 0 0 0 0 0 0 0 1 1 1 1 1F 1 1 1 1 1 0 0 1 1 1 1 1G 1 1 0 0 1 0 0 0 0 0 0 1H 1 1 1 1 1 0 0 0 0 0 0 1I 1 1 1 1 1 0 0 0 0 0 0 1J 1 1 1 1 1 1 1 0 0 0 0 1K 1 1 1 1 1 1 1 1 1 1 1 1

Table 2: Rows refer to the models de-scribed in the first column. Numbersstand for numbers of occurrences of thetransition required to revisit the orig-inal state. T-invariants are specified incolumns.

T-invariants refer to multi sets of reactions which can be interpreted as sequences of re-actions the triggering of which will not change the system state. All T-invariants foundare shown in Tab. 1. Invariants t1 - t7 refer respectively to triggering single reactions andtheir related reverse counterparts. These t-invariants are usually called trivial invariants.Other invariants (except t12) describe more complicated cycles of electrons inside pho-tosystem II. The most complex is invariant t12 expressing the whole set of reactionsneeded for a sequence of two subsequent unbinding events of QB

-2, e.g., sending 4



electrons to the next photosynthetic system. In Tab. 2, T-invariants are shown mappedto the respective models.

3.4 Analysis by CTL

We employed model checking to study individual dynamic phenomena described interms of formulae of the computation tree logic (CTL) [4]. The most important resultwe have obtained is given in Fig. 5. It explains relations between created models andshows the validity of individual CTL formulae.

Fig. 5: The graphs show an overview of the model checking results of the models de-scribed in Section 3.1. Relation between models X and Y (considering X positionedabove Y) naturally means that X is a refinement of Y (it contains a superset of all com-ponents in X). Generally speaking the models on the top of the graph are the most de-tailed and complex. On the other hand, the models on the bottom are the simplest ones.Transitive and reflexive connections are missing. For simplification, there are missingalso connections between models containing OEC (group a) and models without OEC(group b). Colours of individual circles appearing in nodes refer to validity of respectiveCTL formulae. The formula ϕ = 1 for a model X ⇔ the respective node is coloured bythe color assigned to ϕ.



Formula φ1 describes the possibility of quinone QB reduction that allows unbinding ofQB from the system thus making other parts of photosynthetic reactions possible.

φ1 = EG(EF(P∧QA ∧QB)∧!(QB-2)) (1)

Formally, φ1 requires there exists a path in the reachability graph such that P/Qa/Qbis always reachable and Qb never reaches the twice reduced state. Validity of φ1 meansthat Qb is never left unbound from PSII.

The formula has the same truth value for all corresponding models from both groups.Thus we can assume that the expressed property is independent of OEC function.

The formula is false only for models where the electron can cycle through reversibletransitions without reaching QB

-2. In other words, presence of any other T-invariant thant12 violates the formula.

Formula φ2 verifies if there exists for every state a possibility to reach a state whereP680 is neutralized. This can be interpreted as the possibility to avoid the irreversiblephotosystem damage caused by long time oxidation of P680.

φ2 = AG(P+ → EF(P)) (2)

Formula φ2 is true for all models from group a where P680 can be always reducedby an electron from OEC. For models Ab, Bb a Db the formula is false because there areno reverse transition capable of reducing P680. In terms of T-invariants, the models donot contain any other T-invariant than only t12.

Despite of the fact that models Cb, Gb and Hb enable the excited electron to return,a specific state can be reached where the electron cannot be returned back. If is P680after first excitation reduced by electron from YZ and this electron is also excited andtransited to quinone QB, both of these electrons can be taken away and P680 will stayoxidised. Model Eb (Ib) contains in addition cyt b559 (tyrosine YD) capable of donatingone electron to the system. The number of electrons is therefore 3 and although twoof them can leave the system, one will be always able to cycle and periodically reduceP680.

Interesting is model Fb which contains both cyt b559 and tyrosine YD. Thus doubleoxidation of QB can be performed two times which would lead to exhaustion of systemand permanently oxidised P680.

Formula is also true for models Jb and Kb where it is always possible to reduce P680by a sequence of reverse transitions even from the PQ-pool.

Formula φ3 describes cyclic behaviour of P680. Verifying the assumption that P680 isneutralized or oxidized, it will always get to the state where is oxidised or neutralized,respectively. In other words, the property ensures that the system does not stack withP680 permanently oxidised or neutralized which would lead to its disfunction causedby the absence of electron excitation events.

φ3 = AG((P → AF(P+)∧ (P+ → AF(P))) (3)

The formula is false for models containing any other reverse reaction than the onerelated with complex P680, in other words, it is false for all models containing any



t-invariant not related with P680. System in these reaction can cycle. This holds alsofor group b. Moreover this formula is true only in case when φ2 holds, i.e., a final statesatisfying P680 cannot be never reduced can be reached. Only model Eb does not haveany final state and thus the formula is true. The reason is that the model contain onlyreverse reactions related to P680 ensuring its periodic reduction.

Formula φ4 expresses the condition that if P680 is oxidised and QA and QB are neutral-ized, there is only one possible action - neutralization of P680+.

φ4a = AG((P+ ∧QA ∧QB)→ AX((P∧QA ∧QB))) (4)

To make the property clearer we define another formula φ4a varying for different modelsin order to ensure that all oxidizable (reducible) components are oxidized (reduced), re-spectively. This state displays an absolute lack of electrons in the system. For example,for the model including molecules YZ and YD the formula has the following form:

φ4b =AG((P+ ∧QA ∧QB ∧YD+ ∧YZ

+)→AX((P+ ∧QA ∧QB ∧YD

+ ∧YZ)))(5)

Formula φ4a is true only for basic models Aa and Ab. In other models, there is includedsome other component capable of another step than reducing oxidised P680. More spe-cific formula φ4b ensures that it will not happen. It is false only for models Ja,Ka,Jb andKb which contains t-invariant t7 that means unbinding of neutral quinone QB (QB → E).

3.5 Discussion

Petri nets showed up as a feasible modelling formalism suitable for our problem. Allmodels, even the most complex ones, are simple and transparent.

Considered binary interpretation of places and transitions seems to be optimal, be-cause other interpretations would not be so explicit about states of components and itwould be necessary to define maximal number of tokens present in every place.

T-invariants provided a simple way for representation and specification of non-trivial transitions needed for revisiting the initial state.

Formulas used for expressing observed features were also very transparent and eas-ily understandable. Nevertheless, they proved some behaviour of system is possibledespite it is actually very improbable, even impossible. An example is infinite cyclingof electron using a forward reaction and its respective reverse direction.

An interesting result is that some formulas have the same truth value for corre-sponding model variants (a) and (b). Thus we can assume that expressed features donot dependent on the functionality of OEC. A special attention should be paid to modelswithout OEC because some of them, especially the very comprehensive model Fb, canreach critical state with permanently oxidised P680. Special attention is needed duringformulas formulation, e.g., formula φ4a. It is valid only for the simplest models in thehierarchy. An alternative formula φ4b has to be employed for other models.

Details on models, formulas and (im)possibility of verifying the requested featuresshould be an objective of further discussions with scientists interested in photosystem II.



4 Conclusions and Future Work

In this paper, known facts about photosystem II have been summarized and we haveshown preliminary results on its qualitative modelling and analysis. Created modelshave been compared wrt validity of CTL formulas representing crucial features of thephotosystem. It has been shown that some features do not depend on the functionalityof OEC. Moreover, it has been shown that some models can reach the final state inwhich the complex P680 is oxidised. Such a scenario would lead to disfunction andirreversible damage of photosystem II.

For future work we consider addition of formulas reflecting other features of thesystem. Furthermore, our research aims at targeting other protein complexes in the pho-tosynthesis chain. Their models can be analysed individually and also integrated intoa complex photosynthesis model. Obtained information about models can be also val-idated in a real system (if it is possible) or compared to existing quantitative models.Other direction could be an implementation of software capable of creation and analysisof all 4320 possible models of photosystem II and visualization of validity of specifiedformulas in similar way as in the graph shown in Fig. 5.

References

1. Barber, J., Rivas, J.D.L.: A functional model for the role of cytochrome b559 in the protectionagainst donor and acceptor side photoinhibition. Biochemistry 90(23), 42– 46 (1993)

2. Belyaeva, N.E., Pahchenko, V.Z., Renger, G., Riznichenko, G.Y., Rubin, A.B.: Applicationof a photosystem II model for analysis of fluorescence induction curves in the 100 ns to 10 stime domain after excitation with a saturating light pulse. Biophysics 51, 860–872 (2006)

3. Bonzanni, N., Krepska, E., Feenstra, K.A., Fokkink, W., Kielmann, T., Bal, H., Heringa,J.: Executing multicellular differentiation: Quantitative predictive modelling of C.elegansvulval development. Bioinformatics 25(16), 2049–2056 (2009)

4. Clarke, E.M., Emerson, E.A., Sistla, A.P.: Automatic verification of finite-state concurrentsystems using temporal logic specifications. ACM Trans. Program. Lang. Syst. 8(2), 244–263 (Apr 1986), http://doi.acm.org/10.1145/5397.5399

5. Franzke, A.: Charlie 2.0 – a multi-threaded Petri net analyzer. Diploma thesis, BTU Cottbus,Dep. of CS (December 2009)

6. Gilbert, D., Heiner, M.: From petri nets to differential equations - an integrative approach forbiochemical network analysis. In: ICATPN’06. pp. 181–200. LNCS, Springer (2006)

7. Hardy, S., Robillard, P.N.: Pn: Modeling and simulation of molecular biology systems usingpetri nets: modeling goals of various approaches. J Bioinform Comput Biol 2004, 2–4 (2004)

8. Heiner, M., Gilbert, D., Donaldson, R.: Petri nets for systems and synthetic biology. In:Proceedings of the Formal methods for the design of computer, communication, and softwaresystems 8th international conference on Formal methods for computational systems biology.pp. 215–264. SFM’08, Springer (2008)

9. Lazar, D.: Modelling of light-induced chlorophyll fluorescence rise (O-J-I-P transient) andchanges in 820 nm-transmittance signal of photosynthesis. Photosynthetica 47, 483–498(2009)

10. Lazar, D., Jablonsky, J.: On the approaches applied in formulation of a kinetic model ofphotosystem II: Different approaches lead to different simulations of the chlorophyll a fluo-rescence transients. J Theor Biol. 257 (2009)

11. Lazar, D., Schansker, G.: Models of chlorophyll a fluorescence transients. In: Photosynthesisin silico: understanding complexity from molecules to ecosystems, Advances in Photosyn-thesis and Respiration, vol. 29, pp. 85–123. Springer (2009)



12. Litvın, R.: Photochemistry of photosystem II reaction centre: Alternative electron pathways.Ph.D. thesis, University of South Bohemia, Faculty of Science (2009)

13. Lokstein, H., Hrtel, H., Hoffmann, P., Woitke, P., Renger, G.: The role of light-harvestingcomplexii in excessexcitationenergy dissipation: An in-vivofluorescencestudy onthe origin of high-energyquenching. Journal of Photochemistry and Photobiology 26(2),175–184 (1994)

14. Manca, V., Pagliarini, R., Zorzan, S.: A photosynthetic process modelled by a metabolic psystem. Natural Computing 8(4), 847–864 (Dec 2009)

15. Nedbal, L., vCerveny, J., Schmidt, H.: Scaling and integration of kinetic models of photosyn-thesis: Towards comprehensive e-photosynthesis. In: Laisk, A., Nedbal, L., Govindjee (eds.)Photosynthesis in silico: understanding complexity from molecules to ecosystems, Advancesin Photosynthesis and Respiration, vol. 29, pp. 17–29. Springer (2009)

16. Pedersen, M.: Compositional definitions of minimal flows in petri nets. In: Proceedings of the6th International Conference on Computational Methods in Systems Biology. pp. 288–307.CMSB ’08, Springer-Verlag (2008)

17. Rappaport, F., Guergova-Kuras, M., Nixon, P.J., Diner, B.A., Lavergne, J.: Kinetics and path-ways of charge recombination in photosystem II. Biochemistry 41(26), 8518–8527 (2002)

18. Reddy, V.N., Liebman, M.N., Mavrovouniotis, M.L.: Qualitative analysis of biochemicalreaction systems. Computers in Biology and Medicine 26(1), 9 – 24 (1996)

19. Rohr, C., Marwan, W., Heiner, M.: Snoopy–a unifying petri net framework to investigatebiomolecular networks. Bioinformatics 26(7), 974–975 (2010)

20. Ruffle, S., Sayre, R.: Functional analysis of photosystem II. In: Rochaix, J., Goldschmidt-Clermont, M., Merchant, S. (eds.) The Molecular Biology of Chloroplasts and Mitochon-dria in Chlamydomonas, Advances in Photosynthesis and Respiration, vol. 7, pp. 287–322.Springer (2004)

21. Schaub, M., Henzinger, T., Fisher, J.: Qualitative networks: a symbolic approach to analyzebiological signaling networks. BMC Systems Biology 1(1), 1–21 (2007), http://dx.doi.org/10.1186/1752-0509-1-4

22. Shinopoulos, K.E., Brudvig, G.W.: Cytochrome b559 and cyclic electron transfer withinphotosystem II. Biochimica et Biophysica Acta (BBA) - Bioenergetics 1817(1), 66 – 75(2012)

23. Zhu, X.G., Govindjee, Baker, N., deSturler, E., Ort, D., Long, S.: Chlorophyll a fluorescenceinduction kinetics in leaves predicted from a model describing each discrete step of excitationenergy and electron transfer associated with photosystem II. Planta 223, 114–133 (2005)


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